Math Identity Prover & Verifier

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Category Math & Algebra

Enter Identity (LHS = RHS)

Common Identity Library

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About

Mathematical verification is the backbone of rigorous analysis in engineering, physics, and higher education. Errors in algebraic manipulation or trigonometric substitution often lead to cascading failures in complex derivations. This tool serves students, educators, and engineers who need to validate the equality of two mathematical expressions with absolute certainty.

Unlike standard calculators that merely output a numerical result, this Identity Prover establishes logical equivalence. The engine employs a hybrid architecture: a Symbolic Pattern Matcher checks against a library of over 500 standard curriculum identities, while a Numerical Analysis Engine verifies custom expressions by evaluating them across the complex plane. This dual approach detects subtle errors in sign, coefficient magnitude, and functional application that simple parsers miss.

Use this tool to check homework steps, verify intermediate results in calculus problems, or explore relationships between trigonometric functions. The system rigorously handles standard trigonometric identities (Pythagorean, Double Angle, Sum-to-Product) and complex algebraic expansions.

Formulas

The verification engine relies on fundamental equivalence principles. For trigonometric analysis, the Pythagorean Identity is the baseline for simplification:

sin²(x) + cos²(x) = 1

Algebraic verification often requires the expansion of polynomials to compare coefficients. The generalized binomial expansion for degree 2 is:

(x + y)² = x² + 2xy + y²

For complex analysis involving exponentials and trigonometry, the engine references Euler's Identity:

e^iπ + 1 = 0

Reference Data

Category	Identity Name	Expression (Standard Form)
Trigonometry	Pythagorean Identity	sin²(x) + cos²(x) = 1
Trigonometry	Double Angle (Sine)	sin(2x) = 2sin(x)cos(x)
Trigonometry	Double Angle (Cosine)	cos(2x) = cos²(x) − sin²(x)
Algebra	Difference of Squares	a² − b² = (a − b)(a + b)
Algebra	Perfect Square Trinomial	(a + b)² = a² + 2ab + b²
Algebra	Sum of Cubes	a³ + b³ = (a + b)(a² − ab + b²)
Trigonometry	Tangent Identity	tan(x) = sin(x)cos(x)
Trigonometry	Cotangent Identity	cot(x) = cos(x)sin(x)
Calculus	Euler's Formula	e^ix = cos(x) + isin(x)
Logarithms	Product Rule	log_b(xy) = log_b(x) + log_b(y)
Logarithms	Power Rule	log_b(xⁿ) = n log_b(x)

Frequently Asked Questions

No. This tool specifically targets algebraic and trigonometric identities. It does not perform symbolic differentiation (d/dx) or integration (∫). It is designed to verify static equalities, such as simplifying a trigonometric expression to a constant or checking a factorization.

The engine uses a two-tier system. If the input matches a known pattern in the symbolic library (e.g., standard identities), it provides a formal step-by-step proof. For custom or highly complex expressions not in the library, it performs a rigorous numerical check by evaluating both sides at multiple random points in the complex plane. This confirms equality with high probability but does not generate logical text steps.

Use standard mathematical notation. You can write "sin(x)^2" or "(sin(x))^2". The parser automatically interprets these as the square of the sine function. Avoid writing "sin^2(x)" in the raw text input if possible, as strict parsers prefer "sin(x)^2", though this tool attempts to auto-correct common formatting variances.

Yes. The verifier supports standard variables like x, y, a, b, theta, and phi. When using multiple variables, the numerical verification engine ensures the identity holds true across a multi-dimensional sample space.

An identity is true for ALL values of the variable (e.g., x + x = 2x). An equation is only true for specific values (e.g., x + 2 = 5, true only if x=3). This tool is built to detect identities; if you enter a conditional equation, it will likely return "False" because it is not true for all random test points.